Types of narrators are described by literary critics in terms of the pronouns used by the narrator and the types of knowledge to which the narrator has access. In the case of A Dog's Tale by Mark Twain, the narrator uses the word "I", which is a first person pronoun, and thus we describe the story as being written in the first person, having a first-person narrator, or written from a first-person point of view.
A Dog's Tale begins with the line:
My father was a St. Bernard, my mother was a collie, but I am a Presbyterian.
This tells us that the narrator is a dog, but one who is anthropomorphic in some aspects, as normally one does not associate dogs with religious denominations. The narrator explains this affiliation in terms of his mother's intellectual interests. The dog tells us that her name is Aileen Mavourneen. As the story progresses, we get an account of vivisection as it appears from the dog's point of view. The choice of narrator is Twain's attempt to make us empathize with animals by showing us the horrors of vivisection (to which he was strongly opposed) by imagining how it would appear through the limited understanding of a dog.